Localized waves along a line of masses on a plate: propagation and sub-exponential attenuation

Chakraborty, Ishita and Mohanty, Atanu K and Chatterjee, Anindya
(2008)
Localized waves along a line of masses on a plate: propagation and sub-exponential attenuation.
In: Proceedings of the Royal Society A- Mathematical, Physical and Engineering Sciences, 464
(2097).
pp. 2229-2246.

Abstract

We have studied waves propagating in an infinite plate with a line of equally spaced point masses on it. In particular, we consider waves that are laterally decaying, i.e. waves that exist due to the point masses, not in spite of them. We begin with a simple continuum limit and then study the discrete case using a suitable Green function and a superposition method used earlier for other structures by Mead. The superposition involves a slowly decaying series that we transform to a rapidly converging one using the Poisson summation formula. The system has two non-dimensional parameters, and on this parameter plane we find a propagation zone and its boundary. The boundary involves successive point masses vibrating n radians out of phase, a situation far from the continuum limit. Outside the propagation zone, waves get attenuated; however, unlike common examples of linear periodic structures, and contrary to the usual assumption made in studies thereof, here the waves show sub-exponential attenuation. In particular, the eventual decay of the wave amplitude is like the 3/2 power of distance from the point of excitation. We conclude with a theoretical explanation of the 3/2 power in the decay.